Introduction

Assessing the water clarity of waterbodies is a key issue for environmental monitoring and management. The Secchi depth (SD) is an optical measure of water clarity by human vision and it is determined by all optically active substances (OASs) in the water, i.e., phytoplankton (represented by chlorophyll-a), tripton [non-planktonic suspended solids (SSs): Tr], colored dissolved organic matter, and pure water (Mancino et al., 2009). Despite the ease with which SD data can be obtained, the collection of these data frequently and for huge numbers or for large waterbodies is costly and challenging for monitoring agencies (Nelson et al., 2003). Remote sensing offers several advantages such as spatially and temporally extensive coverage and the possibility of measuring many waterbodies simultaneously (Koponen et al., 2002). Therefore, a combination of remote-sensing techniques with precise knowledge on water optics could provide an efficient method to monitor waterbodies.

Many research groups have investigated the relationships between in situ measurements of SD and the spectral response of satellite sensors such as Landsat (Lathrop et al., 1991; Giardino et al., 2001; Kloiber et al., 2002; Nelson et al., 2003; Sawaya et al., 2003; Chipman et al., 2004; Hellweger et al., 2004; Alparslan et al., 2007; Sass et al., 2007; Olmanson et al., 2008; Kabbara et al., 2008; Mancino et al., 2009; Hicks et al., 2013; Olmanson et al., 2014), MODIS (Härmä et al., 2001; Wu et al., 2008), SeaWiFS (Kratzer et al., 2003; Binding et al., 2007), and MERIS (Härmä et al., 2001; Koponen et al., 2002; Kratzer et al., 2008; Giardino et al., 2010). However, most of these studies focused principally on empirical approaches, and thus the final applications tended to be time- and site-specific. For example, if an algorithm has been developed for an algae-dominant water body, it is difficult or impossible to apply this algorithm to a non-algae-dominant water body, because different OASs in the water bodies will result in different bands selection and thus different algorithms. Thus, for example, the SD prediction scheme through chlorophyll-a (Chl-a) possibly results in fairly large estimation errors.

In parallel, substantial efforts have been made to retrieve the inherent optical properties (IOPs) of waterbodies from spectral measurement of water color. These efforts have included an empirical algorithm (Lee et al., 1998), a spectral optimization approach (Roesler & Perry, 1995), an artificial neural network (Doerffer & Schiller, 2007), a linear matrix inversion method (Hoge & Lyon, 1996), and a quasi-analytical algorithm (QAA; Lee et al., 2002). Among these approaches, the QAA, which is an inversion based on the semi-analytical relationship between remote-sensing reflectance and the IOPs, is quite effective and simple to implement. Chen et al. (2007) estimated the SD in Tampa Bay, Florida by processing SeaWiFS satellite imaginary based on a two-step process, first estimating the diffuse attenuation coefficient at 490 nm [K d(490)] using this semi-analytical approach, and then the SD using as empirical relationship with K d(490). This method has provided an excellent estimate of in situ SD values, but its applicability is uncertain for wider SD ranges of waterbodies because of both the empirical relationship used with K d(490) estimation and the far lesser reliability of the semi-analytical approach for inland waters, particularly turbid and eutrophic waters.

There is another approach—a semi-analytical approach that uses the contrast transmittance theory (Tyler, 1968; Preisendorfer, 1986). In this approach, SD values are expressed using the beam and diffuse attenuation coefficients in the range of visible wavelengths, which could be related to the IOPs (i.e., the absorption and scattering coefficients) of the targeted water. This approach has seldom been used to estimate SDs in lakes because the measurement of attenuation coefficients and/or IOPs requires a high-level of technique and a great deal of labor in the field or laboratory; a similar trial has been done for ocean transparency using MERIS, MODIS, and SeaWiFS data, however, showing substantial prediction errors (Doron et al., 2011). However, it may be possible to use the IOP-retrieving algorithm from remote-sensing reflectance, as mentioned above.

In this approach, IOPs are estimated using remote-sensing reflectance spectra at the first stage, and then the values of SD are predicted using the contrast transmittance theory at the second stage. Because both stages are based on semi-analytical algorithms, this approach is expected to be applicable to wider water regions compared to the method based on empirical relationships. To establish this approach, an algorithm for retrieving IOPs in turbid waters is absolutely necessary.

Yang et al. (2013) proposed a modified algorithm (enhanced QAA for turbid waters: QAA_turbid) to retrieve total absorption and backscattering coefficients based on a semi-analytical estimation model for turbid inland waters and they described the potential of this algorithm to accurately retrieve the IOPs using in situ reflectance spectra. A hybrid scheme could thus be constructed for the estimation of IOPs: original QAA for clear waters and QAA_turbid for turbid waters. However, a classification procedure is necessary for choosing which IOPs estimation method should be applied.

In the present study, we constructed a semi-analytical algorithm for estimating SD value using remote-sensing reflectance spectrum. We then compared the estimated SD values with the measured values that were obtained in 10 Japanese lakes with a wide variety of turbidity. We propose a classification procedure between clear and turbid states and the coefficient in the contrast transmittance theory. Then, an accuracy comparison with the results obtained by previous studies is made. The coefficients used in the proposed system and discrepancies between the measured and predicted SDs are discussed in addition to the future directions for this approach. In this study, we use in situ-measured reflectance spectra in the range of 400–780 nm at 1-nm intervals. The application of the present approach to satellite data is a future challenge.

Methods

In situ data collection

Measurements of SDs and corresponding reflectance spectra, and collections of water samples were done at 10 lakes in Japan (Table 1; Fig. 1) during the years 2009–2014. Fifteen data collection campaigns were used for this study: four campaigns in Lake Kasumigaura, two campaigns in Lakes Biwa and Shoji, and one campaign for the other seven lakes. The numbers of the observation points at the respective campaigns are shown in Table 1.

Table 1 The 10 surveyed lakes in Japan and their dimensions, sampling information, and water quality
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Fig. 1
figure 1

The location of the 10 lakes in Japan

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SDs were measured using a standard white circular panel (0.3 m in dia.). Reflectance measurements were performed between 9:30 and 16:00 h local time. The upwelling radiance (L u), the downward irradiance (E d, reflected upwards from the SRT-99-100 Spectralon reflectance panel®), the downward radiance of skylight (L sky), and the upwelling radiance from the reflectance panel shaded from direct sunlight (L shade) were measured at each site using a FieldSpec HandHeld spectroradiometer (Analytical Spectral Devices, Boulder, CO, USA) in the range of 325–1075 nm at 1-nm intervals. “Lsky” refers to the diffused radiation of the sky, which contains no information on water properties, and hence must be eliminated. To avoid the influence of reflection and shading from the boat, all spectral observations were measured at the zenith angle of around 58°. In addition, the direction of each spectral observation was 90° to the plane of the sun to avoid the effect of sun glint.

The above-water remote-sensing reflectance (R rs) was calculated approximately using the equation given by Mobley (1999):

$$R_{\text{rs}} \approx \left( {\frac{{L_{\text{u}} }}{{E_{\text{d}} }} - \frac{{F(\theta ) \cdot L_{\text{sky}} }}{{E_{\text{d}} }}} \right) \cdot {\text{Cal}} \cdot 100$$
(1)

where Cal is the spectral reflectance calibration factor for the Spectralon reflectance panel (SRT-99-100). F(θ) represents the reflectance of the skylight at the air–water interface ranging from 0.022 to 0.025, and θ is the measurement view angle (Mobley, 1999). In this study, a constant value of 0.024 was used for F(θ).

At in situ measurements, the surface waves and clouds affected the obtained reflectance spectra. We thus removed the campaigns when the influence of waves seemed considerable. The upwelling radiances were measured 10 times, and then the lower five data were averaged to get the value of L u. Here, we calculated the coefficient of variation (CV; the standard deviation divided by the average) of the lower five radiances averaged in the range of 400–750 nm and considered the influence of waves insignificant if the coefficient was <0.07. All of the campaigns shown in Table 1 satisfied this condition (more than 10 campaigns not shown in Table 1 were unavailable). The influence of waves surely relates to wind velocity; for example, the maximum wind velocities (3.3, 4.5 and 3.9 m/s at Hikone, Nagahama, and Imazu near the lake, respectively) during the excluded campaign in Lake Biwa (6 September 2010) were higher than those during the included campaigns (2.6, 2.2, and 2.0 m/s on 17 August 2009 and 2.2, 2.0, and 2.2 m/s on 12 October 2011 at those stations, respectively) (Japan Meteorological Agency, 2015).

Next, the influence of clouds was considered using the value of L shade divided by (E d/π). For the subsequent calculation, we used the measurement when this value was <0.25. The numbers of measurements that satisfied this condition are shown in Table 1 as the number of points. In addition, the threshold was changed to 0.20 to evaluate the influence of clouds for estimating SDs. The R rs spectra used for the further analysis are shown in Fig. 2.

Fig. 2
figure 2

In situ reflectance spectra used in this study. a Lake Kasumigaura, b Lake Biwa, c Lake Suwa and Lake Shirakaba, d other lakes

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For the collected water samples, Chl-a was extracted using methanol (100%) at 4°C under dark conditions for 24 h. The extract was then centrifuged at 3,000 rpm for 10 min and analyzed spectrophotometrically to estimate the concentration of Chl-a according to SCOR-UNESCO equations (1966). The concentrations of the SSs were determined gravimetrically. Samples were filtered through pre-combusted Whatman GF/F filters at 500°C for 4 h to remove the dissolved organic matter in suspension, which was then dried at 105°C for 4 h and weighed to obtain SS. Table 1 provides a summary of the relevant characteristics of the studied waterbodies together with the SDs and OASs (Chl-a and SS) concentration ranges obtained from the in situ data collection.

Development of a semi-analytical model for estimating SDs

The flow chart of the following proposed method is shown in Fig. 3.

Fig. 3
figure 3

Flow chart of the semi-analytical prediction of Secchi depth. R rs remote-sensing reflectance, MCI maximum chlorophyll index, QAA_v5 quasi-analytical algorithm version 5, QAA_turbid quasi-analytical algorithm for turbid waters, a absorption coefficient, b b backscattering coefficient, c beam attenuation coefficient, K d diffuse attenuation coefficient, Γ coupling constant, SD Secchi depth

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Stage 1: retrieving IOPs from reflectance spectra

As explained below, two algorithms for retrieving IOPs from reflectance spectra were prepared for clear and turbid waters, respectively. Since a selection between them is therefore necessary, we used the maximum Chl index [MCI: R rs(709 nm) − R rs(681 nm)], which is a measure of the reflectance peak at 709 nm in water-leaving reflectance. Gower et al. (2005) have demonstrated use of this index to detect plankton blooms. This is because the absorption by Chl reduces radiance at wavelengths shorter than 700 nm, while absorption by water reduces radiance at wavelengths longer than 720 nm, leading to a radiance peak at the wavelength of minimum absorption, near 709 nm. This index was used to select the most appropriate algorithm for estimating Chl-a concentrations among a blue–green algorithm, an algorithm with a two-band index, and an algorithm with a three-band index (Matsushita et al., 2015). In this study, three ranges were set for the selection of Chl-a estimating algorithms: (1) when the MCI was ≤0.0001, then the blue–green algorithm was used, (2) when 0.0001 < MCI ≤ 0.0016, then the two-band algorithm was used, and (3) when MCI >0.0016, then the three-band index algorithm was used. The MCI values 0.0001 and 0.0016 approximately corresponded to the Chl-a values 10 and 25 mg/m3, respectively, based on a linear relationship between MCI and Chl-a (R 2 = 0.76, P < 0.001) using the data taken in five Asian lakes. The results showed that this hybrid algorithm performed well in these lakes (Lakes Biwa, Kasumigaura, and Suwa used in this study and Lakes Dianchi and Erhai in China not used in this study but explained in Matsushita et al., 2015).

Following the original proposal of the QAA by Lee et al. (2002), several updates have been presented (Lee et al., 2005, 2007). For clear waters, we used a recent version (version 5, denoted QAA_v5 hereafter; Lee et al., 2009) in which four empirical estimation models have been updated compared to previous versions. The flow of QAA_v5 is shown in Table 2, and the steps up to Step 6 were used in this study. There are mainly three limitations that challenge the applicability of QAA_v5 in turbid inland waters. The first limitation regards about the estimation model for total absorption at the reference band (Step 2 in Table 2), which is empirically derived from the synthetic dataset. The second limitation regards the Y estimation model (Step 4), which was calibrated using the data obtained in oceans or coastal areas. The third limitation is the algebraic equations for estimating the phytoplankton absorption (Steps 7 and 8), in which the ratios were empirically determined in a manner similar to that of Step 4. Yang et al. (2013) proposed the enhancement of the QAA for turbid waters in three ways: (1) the extension of the reference wavelength to the near-infrared band, (2) the development of a semi-analytical estimation model for the spectral slope of particle backscattering, and (3) the development of an empirical estimation model for the phytoplankton absorption coefficient at 443 nm. The main steps are shown in Table 3 (denoted “QAA_turbid” hereafter) and the steps up to Step 5 were used in the present study.

Table 2 Steps of the quasi-analytical algorithm for retrieving IOPs in clear waters: QAA_v5 (Lee et al., 2009)
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Table 3 Steps of the quasi-analytical algorithm for retrieving IOPs in turbid waters: QAA_turbid (Yang et al., 2013)
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Thus, we applied two MCI thresholds for the selection of IOP-retrieving algorithms, i.e., (1) when MCI ≤0.0001, then QAA_v5 and (2) when MCI >0.0016, then QAA_turbid. In addition, both algorithms were compared in the range of 0.0001 < MCI ≤ 0.0016.

Stage 2: estimating SDs from IOPs

Based on the constant transmittance theory (Tyler, 1968; Preisendorfer, 1986), the following SD relation was developed.

$${\text{SD}} = \frac{\varGamma }{{\int_{{\lambda_{1} }}^{{\lambda_{2} }} {{\text{Eye}}(\lambda ) \cdot I_{\text{surface}} (\lambda )[{\user2{c}}(\lambda ) + {\user2{K}}_{\text{d}} (\lambda )]{\text{d}}\lambda /\int_{{\lambda_{1} }}^{{\lambda_{2} }} {{\text{Eye}}(\lambda ) \cdot I_{\text{surface}} (\lambda ){\text{d}}\lambda } } }}$$
(2)

where c and K d are depth-averaged (from the water surface to the SD) beam and diffuse attenuation coefficients, respectively. When these coefficients do not change vertically, they are expressed simply as c and K d, respectively. Hereafter, we assume their vertical constancy. Eye(λ) and I surface(λ) are the photopic response of the human eye and the downwelling irradiance at the water surface, respectively, and they are a function of the wavelength: λ. The denominator of Eq. (2) indicates the sum of c and K d weighted toward the wavelength dependency of the photic response and the downwelling irradiance over the range of visible wavelengths from λ 1 to λ 2 (Preisendorfer, 1986). The photopic function possesses a maximum of around 550 nm and near zero at the two ends λ 1 and λ 2 (Takami, 2011), as shown in Fig. 4a. In this study, λ 1 and λ 2 were set at 400 and 750 nm, respectively. We used the smoothed “global tilt” (reference solar spectral irradiance: ASTM G-173) as the downwelling irradiance at the water surface as shown in Fig. 4a.

Fig. 4
figure 4

Optical properties used in this study. a The photopic response of the human eye (broken line) and the downwelling irradiance at the water surface (solid line). b The absorption coefficient (solid line) and backscattering coefficient of pure water (broken line)

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In addition, Γ is a coupling constant that depends on variations in ambient conditions during measurements (Preisendorfer, 1986). In the present study, the value of Γ was set at (1) 9.7 in all lakes (hereafter indicated as Case 1), (2) 9.7 in turbid lakes and 8.2 in clear lakes (indicated as Case 2), and (3) using the following Eq. (3) (indicated as Case 3, but as Case 4 when using the threshold of cloud influence [L shade/(E d/π)) = 0.20].

$$\varGamma = 8.12 \times \left( {\int_{{\lambda_{1} }}^{{\lambda_{2} }} {{\text{Eye}}(\lambda ) \cdot I_{\text{surface}} (\lambda )[c(\lambda ) + K_{\text{d}} (\lambda )]{\text{d}}\lambda /\int_{{\lambda_{1} }}^{{\lambda_{2} }} {{\text{Eye}}(\lambda ) \cdot I_{\text{surface}} (\lambda ){\text{d}}\lambda } } } \right)^{0.1076}$$
(3)

The values of (1) and (2) were obtained by the statistical analysis in Terrel et al. (2012). Equation (3) was deduced from the regression analysis between SD and (c + K d) (Fig. 5) using the data from Davies-Colley (1988) (eight lakes with the range of SD from 0.42 to 17.7 m). Davies-Colley considered that this type of tendency resulted from the change in the angle subtended by the target disk at the eye with SD.

Fig. 5
figure 5

Relationship between (c + K d) and SD using the data by Davies-Colley (1988). Solid line the regression line. Dotted line the line of the negative first power of (c + K d)

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The beam attenuation coefficient c(λ) is expressed as the sum of the absorption and scattering coefficients:

$$c(\lambda ) = a(\lambda ) + b(\lambda )$$
(4)

Absorption a(λ) and scattering b(λ) are IOPs, and thus can be expressed as the sum of the contributions, e.g., pure water, phytoplankton, tripton, and colored dissolved organic matter.

The diffuse attenuation coefficient K d(λ) is estimated based on the relationship given by Kirk (1984):

$$K_{\text{d}} (\lambda ) = \left[{a(\lambda)^{2} + k_{\mu} \cdot a(\lambda ) \cdot b(\lambda)} \right]^{0.5}/ \mu_{0}$$
(5)

The coefficient k μ was estimated from the following equation (Bowers et al., 2000):

$$k_{\mu } = 0.425 \cdot \mu_{0} - 0.19$$
(6)

where μ 0 is the cosine of the zenith angle of refracted solar photons, calculated from the solar zenith angle (SZA) using Snell’s law with the appropriate index of refraction for water and air. The SZA was calculated by feeding the date, time, and latitude into the algorithm provided by Nakagawa (2015). In addition, Stage 1 gives the value of backscattering of particles: b bp, and then we used the following equation for calculating the value of b:

$$b = b_{\text{bp}} \times {\text{coef}}_{\text{bp}} + b_{\text{bw}} \times {\text{coef}}_{\text{w}}$$
(7)

where b bw is backscattering by pure water (Morel, 1974) as shown in Fig. 4b. In the present study, coefbp was set at 55.6 based on the results of the studies such as Mobley et al. (1993) and coefw was set at 2 due to isotropic scattering. When checking the performance of the present method, we used the absorption coefficient of pure water as shown in Fig. 4b (a w: Pope & Fry, 1997 from 400 to 700 nm and Hale & Querry, 1973 above 700 nm) as the value of a and considered b bp = 0. Then, the SD of pure water was calculated to be 34.8 m.

Accuracy assessment

We used the root-mean-square error (RMSE), the mean normalized bias (MNB), and the normalized RMSE (NRMS) to assess the model performance, defined as follows:

$${\text{RMSE}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{N} {(X_{{{\text{pred}},{\text{i}}}} - X_{{{\text{meas}},{\text{i}}}} )^{2} } }}{N}}$$
(8)
$${\text{MNB}}\;(\% ) = \bar{\varepsilon } = {\frac{{\sum\nolimits_{i = 1}^{N} {\varepsilon_{i} } }}{N}}$$
(9)
$${\text{NRMS}}\;(\% ) = \sqrt {\frac{{\sum\nolimits_{i = 1}^{N} {(\varepsilon_{i} - \bar{\varepsilon })^{2} } }}{N}}$$
(10)

where X pred,i and X meas,i are the predicted and measured values, respectively, N is the number of measurements, and \(\varepsilon_{i} \;(\% ) = 100 \cdot \frac{{(X_{{{\text{pred}},{\text{i}}}} - X_{{{\text{meas}},{\text{i}}}} )}}{{X_{{{\text{meas}},{\text{i}}}} }}\) is the percent difference between the predicted and measured values. The RMSE values depend on the range of SD; in contrast, MNB and NRMS are independent of it, indicating appropriate indicators for evaluating the prediction model in a wide variety of lake light conditions. The standard error (se) of the regression equation (independent variable: measured, dependent variable: predicted) could be calculated as RMSE × ((N − 2)/N)0.5. When comparing the errors in a logarithmic form, we did not calculate MNB and NRMS. This is because the values of ε i in a logarithmic form are meaningless.

Results

The comparison between measured and predicted SDs is summarized in Fig. 6 and Table 4. In order to evaluate the influences of the value of Γ and cloud on prediction accuracy, Cases 1–4 are classified based on these values as explained in “Methods” section. In addition, Cases 4-1 and 4-2 are the special cases which used QAA_v5 and QAA_turbid for all lakes, respectively, and Case 4-3 is corresponding to Case 4 but accessed after logarithmically transformed. The threshold MCI is 0.0016 for all cases. In regards to the value of Γ, better results were obtained by changing it with the range of SDs (Case 2: Fig. 6b and Case 3: Fig. 6c) compared to the constant Γ model (Case 1: Fig. 6a). When we set Γ = 8.2 and 9.7, the values R 2 = 0.925 and 0.925, P < 0.001 and <0.001, RMSE = 1.20 and 1.53 m, MNB = −22.0 and −14.4%, and NRMS = 37.5 and 41.1% were obtained, respectively, indicating worse agreements (figures not shown). In the comparison of the two changing Γ models, Case 3 gave better agreement than Case 2.

Fig. 6
figure 6

Measured SDs versus predicted SDs. a Case 1 in Table 4, b Case 2, c Case 3, d Case 4, and e lakes with SD <1 m (i.e., Lake Kasumigaura) in Case 4 [magnifying the data shown in the circle in (d)]

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Table 4 Summary of the accuracy assessment for predicting SDs
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When we removed seven data obtained in bad light-measurement (by cloud) conditions (Case 4: Fig. 6d), the R 2 (P < 0.001) and RMSE were worse, but the MNB and NRMS were better compared to the original dataset (Case 3). The removal of the largely underestimated data in this bad cloud conditions (0.55/0.31, 0.65/0.24, 0.50/0.35, 0.50/0.32, 0.58/0.29, 0.60/0.27 m, except one overestimation 0.70/0.89 m; measured SD/predicted SD) resulted in improvement of MNB and NRMS.

In all cases from Cases 1 to 4, the values of MNB were negative, which indicated underestimation on the whole. When we separated the results into the group with SD >1 m and the group ≤1 m using the conditions of Case 4, the former showed overestimation (R 2 = 0.838, P < 0.001, RMSE = 1.62 m, MNB = 17.0%, NRMS = 34.3%, and N = 34), while the latter indicated underestimation (R 2 = 0.390, P < 0.001, RMSE = 0.20 m, MNB = − 27.7%, NRMS = 28.3%, and N = 36 as shown in Fig. 6e). In addition, the evaluation in a logarithmic transformation form gave a higher R 2 value (P < 0.001; Case 4-3).

All of the cases shown above used hybrid prediction algorithms with the MCI threshold = 0.0016 (when MCI ≤0.016, then QAA_v5 was used, and when MCI >0.0016, then QAA_turbid was used). We compared the cases with two individual algorithms for retrieving IOPs for clear or turbid waters against all lakes. Both individual algorithms gave worse results (Case 4-1: Fig. 7a only the Lake Kasumigaura data shown by the clear water algorithm and Case 4-2: Fig. 7b by the turbid water algorithm). In particular, the turbid water algorithm sometimes gave quite improbable SDs.

Fig. 7
figure 7

Measured SDs versus predicted SDs. a Lake Kasumigaura by the algorithm for clear waters and Eq. (3) corresponding to Fig. 6e by the algorithm for turbid water. b Lakes other than Lake Kasumigaura by the algorithm for turbid water and Eq. (3) corresponding to Fig. 6d by the algorithm for clear water

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When we changed the threshold MCI to 0.0001, the selection of the retrieving algorithms was altered for nine data (eight for Lake Suwa and one for Lake Akan). The changes in the results were slightly worse (R 2 = 0.915, P < 0.001, RMSE = 1.18 m, MNB = −12.9%, NRMS = 31.1%), but not so large compared to the results of threshold MCI = 0.0016.

Discussion

Comparison with previous SD prediction research

To evaluate the prediction accuracy of our approach, we compared our results with those of nine previous results published in seven papers (Table 5). Härmä et al. (2001) used satellite data simulated based on airborne measurements. Lakes were targeted except Tampa Bay (Chen et al., 2007) and Himmerfjärden, a fjord-like bay (Kratzer et al., 2008). Values of se or RMSE were shown for accuracy assessment except Binding et al. (2007) and Kratzer et al. (2008). Our results have higher regression coefficients (Cases 1–4 in Table 4) and a similar level of se (Case 4-3 in Table 4) compared to those studies, although the studies cited in Table 5 used space-born reflectance data containing atmospheric disturbances except Härmä et al. (2001).

Table 5 Examples of accuracy assessment for predicting SDs in previous studies
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The dynamic range (0.4–17 m) of SD estimation in the present study is much greater than those of the previous studies, which were usually done in limited waterbodies with SD values <10 m. Because the models shown in Table 5 are classified as empirical model based on field data, they are so site-specific that an application of the obtained model to other lakes is rather questionable. The determination of the model parameters is a prerequisite for using the models. In contrast, our prediction scheme is almost analytical and therefore probably applicable to any other lakes in the world. In addition, there is no need to determine the model parameters. In this context, our scheme is promising as one of the universal algorithms for SD prediction. The value of Γ, only one parameter, which possibly seems rather empirical, will be discussed below.

In this study, we used in situ reflectance data at the intervals of 1 nm to investigate the possibility of the semi-analytical prediction of SD for lakes with a wide range of turbidity. In the future, it will be necessary to use the reflectance data at limited satellite bands and check the applicability of the IOP-retrieving algorithm to space-born data. The influence of atmospheric correction should be also evaluated in the process of SD prediction.

In the field, we sometimes observe the variability of observed SD due to differences in observer eyesight and/or observation experience. Here, the precision of SD measurement was evaluated based on the field survey in three lakes by 12 individuals. At the same point in these three lakes, each examiner reported his/her observation on SD successively without information on the other examiner’s reports. The results showed the following CVs: 13.2% for Lake Sohara (average SD = 3.2 m), 15.1% for Bishamon Pond (average SD = 5.5 m), and 5.6% for Lake Onogawa (average SD = 3.5 m), indicating rather large scatter on SD measurements, due probably to eyesight and/or differences in experience among the examiners. In light of these results, the superiority of SD estimation using reflectance measurement (theoretically zero CV) over direct SD measurement could be expected.

Parameters used in this algorithm

There are many reports about the values of Γ. Tyler (1968) indicated Γ = 8.69 semi-theoretically based on C R (the apparent contrast as seen by an observer) = 0.0066 and C 0 (the inherent contrast of a Secchi disk against its background) = 40. Holmes (1970) reported averaged Γ = 9.42 using 13 measurements in Goleta Bay, California. Preisendorfer (1986) showed that Γ is related to a psychophysiological parameter of the human eye–brain system, the reflectance of the medium, and the submerged disk reflectance, and he suggested that its reasonable estimates were found at approx. 8–9. Terrel et al. (2012) reported Γ = 10.1 for Lake Kasumigaura by parameter fitting using long-term data of measured SD and Γ = 9.3 for the same lake using the limited dataset with an IOP analysis. We used Γ = 9.7 for all 10 of the lakes as the first trial (Table 4, Case 1), but it gave quite large overestimation values of SDs for the clear lakes. Then, we set Γ = 8.2 for the clear lakes based on the parameter-fitting result for Lake Biwa (Terrel et al., 2012) similar to the method mentioned above. Nevertheless, Γ = 9.7 was used for the turbid lakes because underestimated results were mostly seen for the turbid lakes even using Γ = 9.7. The results showed better agreement (Table 4, Case 2), but a scientific basis for changing Γ is needed.

Holmes (1970) compared SD values with disks of 20-, 30-, or 50-cm dia. and showed its increasing tendency with diameter. However, the statistical significance of SD differences was denied. In contrast, Davies-Colley (1988) indicated a trend of Ψ with clarity. Ψ is defined as z divided by (c + K d), where z is black-disk visibility (i.e., corresponding to Γ in the case of SD). As the angle subset by the target at the eye decreased with increased range, Ψ apparently also decreased. Davies-Colley gave the equation of Ψ with black-disk visibility. A similar trend is expected for Γ, and thus we determined the relationship as shown in Fig. 5 using the data by Davies-Colley (1988) and then obtained the relation between Γ and (c + K d) as expressed by Eq. (3). This equation gives Γ = 12.87 at SD = 0.2 m, Γ = 10.59 at SD = 1 m, Γ = 9.74 at SD = 2 m, Γ = 8.01 at SD = 10 m, and Γ = 7.37 at SD = 20 m. Since the influence of Γ on SD prediction is considered fairly large for lakes with a wide variety of turbidity, a careful and elaborate measurement on the relationship between Γ and SDs is necessary.

In addition, the maximum SD was calculated to be 34.8 m with the use of the absorption and backscattering coefficients of pure water as mentioned in the “Methods” section. However, for example, Larson et al. (2007) reported the maximum around 41.5 m (June 1997) in Crater Lake, Oregon. Although there is a possibility of champion data obtained by a person with particularly good eyesight, a value of Γ in such a lake and the IOPs should be reexamined together with checks on Eqs. (5)–(7).

Regarding MCI, the data shown in Fig. 7 and Table 4 clearly indicated that the clear and turbid water algorithms should be used for lakes with MCI ≤0.0001 and those with MCI >0.0016, respectively. The problem is the selection of the threshold between 0.0001 and 0.0016. A comparison of predicted SDs between the two algorithms is shown in Fig. 8 for Lakes Suwa and Akan. A better agreement was seen for clear water prediction (i.e., using the threshold MCI = 0.0016), but the observed SD is usually between the two values predicted by the clear and turbid water algorithms. The use of the threshold of MCI = 0.0016 is recommended based on the present study, but more data are necessary to accurately determine the threshold value. In addition, a blended model would be a further challenge to overcome because Moore et al. (2014) successfully used blended retrievals of Chl-a based on an optical water-type classification. As for MCI, Matsushita et al. (2015) indicated a relationship between MCI and Chl-a as mentioned in the “Methods” section.

Fig. 8
figure 8

Comparison of predicted SDs obtained using the clear and turbid water algorithms for Lakes Suwa and Akan, whose MCI values are in the range of 0.0001–0.0016

Full size image

Similarly to MCI, Matthews et al. (2012) proposed a comparison of reflectance peak height between 681 and 709 nm in order to separate oligotrophic/mesotrophic (peak at 681 nm) and eutrophic/hypertrophic waters (peak at 709 nm) using MERIS images. These peaks correspond to Chl-a fluorescence and particulate backscattering, respectively. Using this criterion, we obtained similar judgement results; peaks at 681 nm in Lakes Biwa, Shirakaba, Suwa, Ikeda, Unagi, Motosu, Shoji, and Sai, and peaks at 709 nm in Lakes Kasumigaura and Akan, and thus selection using the MCI is considered general and trustable.

Lastly, an exclusion of data for analysis is considered. A comparison between Cases 3 and 4 indicated that the meteorological conditions (i.e., waves and clouds), affected the SD prediction accuracy. We then investigated the statistical relationship between [L shade divided by (E d/π)] and the absolute value of ε i for the dataset of Case 4 [i.e., (L shade divided by (E d/π)) <0.20], which was, however, judged to be insignificant (R 2 = 0.078, P > 0.05). In any case, a careful in situ measurement of reflectance spectra is needed.

Discrepancies between the measured and predicted SDs

Rather large discrepancies between the measured and predicted SDs were sometimes observed; especially, an underestimation, an overestimation, and scatters were revealed in Lake Kasumigaura (Fig. 6e), Lakes Sai and Shoji (Fig. 6d), and Lake Biwa (Fig. 6d), respectively. The IOP-retrieving algorithm for turbid waters was developed using the data taken in Lake Kasumigaura, and high accuracy was confirmed (Yang et al., 2013). Because the bias of the retrieving algorithm in visible wavelengths was particularly small, other factors would make an underestimation. When applying the method for the discrimination of cyano-dominant waters (SICFpeak <0 and SIPAFpeak >0; Matthews et al., 2012), the presence of cyano-dominance was detected in most of the reflectance spectra in Lake Kasumigaura except for an observation in March. A dominance of cyanobacteria has been observed in this lake from late spring to autumn (Fukushima & Arai, 2015), resulting in horizontally and vertically inhomogeneous patchy distributions of algae. Thus, such an accumulation (i.e., surface scum and patches of cyanobacteria), could probably affect the observation of SD, but the details and the effects on both SD observation and prediction should be investigated. In addition, the prediction scheme for the diffuse attenuation coefficient K d [Eq. (5)] should be reexamined by other schemes for turbid waters (e.g., Yang et al., 2014).

In Lake Biwa, scatters were often observed for noisy reflectance spectra (7 August 2009; Fig. 2b). Noisy spectra were also obtained in Lakes Ikeda and Unagi (Fig. 2d), but good agreement between the measured and predicted SDs was observed. It thus seems unlikely that scatters are attributable to noisy reflectance spectra. In the case of Lakes Shoji and Sai, a relatively large overestimation was obtained. In these lakes, non-uniform vertical distributions of OASs are expected. Odermatt et al. (2012) indicated the importance of the vertical distribution of various parameters in a stratified lake to evaluate the MERIS observations of phytoplankton blooms. Phytoplankton blooms occur in different stratification layers, allowing the assessment of their influence on remote-sensing estimates. Such an influence should be investigated by both modeling and in situ measurements.

Conclusion

In the present study, we constructed a semi-analytical algorithm for estimating SD value using a remote-sensing reflectance spectrum. We compared the predicted SD values with the observed values in 10 Japanese lakes with a wide variety of turbidity. A fairly good agreement between the predicted and observed values was obtained indicating the usefulness of this prediction scheme. Our prediction scheme is almost analytical and therefore applicable to any lake in the world. In this context, our scheme promises to be one of the universal algorithms for SD prediction. However, further investigations are required on the relationships between Γ and SDs, blending algorithm, the influence of algal patchy distribution and/or a vertically non-uniform distribution of OASs on SD observation and prediction, the use of space-born data and more in order to improve the algorithm and its application.